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Friday, January 29, 2016

As we noted, the purpose of Mathematical Statistics is, using the past observations, to evaluate the probability of certain events to be able to predict their occurrence in the future.

Assume, we are dealing with a discrete random variable ξ that can take any value from a set{X1, X2...XN}with corresponding probabilities{P1, P2...PN},but neither values nor probabilities are known to us.

To determine these values and probabilities, that is to determine the distribution of probabilities of random variable ξ, we've made certain number of random experiments (making sure, the conditions of experiments are the same and independent from each other) and observed the values our random variable took.

Let's state up front that to determine these probabilistic characteristics of our random variable precisely in finite amount of time is impossible.

In the Theory of Probabilities part of this course we have defined the probability Pi of a discrete random variable ξ to take specific value Xi as a limit of its frequency of taking this value as the number of experiments tends to infinity.

Therefore, measuring this frequency during some finite time, making only finite number of experiments, is possible only approximately. However, we might think that, as the number of experiments increases, the precision of our approximation is getting better and better.

In most cases this is true, also, strictly speaking, we have to define quantitatively the quality of approximation and agree when to call it sufficient for our purposes.

Tuesday, January 26, 2016

As we noted, the purpose of Mathematical Statistics is, using the past observations, to evaluate the probability of certain events to be able to predict their occurrence in the future.Is it really possible?

Yes, sometimes. But there are certain provisions that must be satisfied to succeed in this endeavor.

The first and most important condition for applying statistical approach to predict the future is stability.Imagine, you want to evaluate the chances of getting two aces from a dealer in a casino. You are not familiar with combinatorics and cannot calculate these probabilities and, instead, you decide to evaluate your chances statistically. So, you sit at the table and ask a dealer to give you two cards. You do or don't get two aces and continue the process. Eventually, after the whole stack of cards (say, 5 full decks of 52 cards each - 260 cards, that is 130 pairs) is exhausted, you get the final result of the number of times you've got two aces. Say, 3 times out of 130.Would you say that the statistically evaluated probability of getting a pair of aces is somewhat around 3/130 (that is, 2.3%)?Absolutely NOT!And the main reason is - the conditions of our random experiment are changing. We started with a set of 5 full decks of cards shuffled together. After the first top two cards are dealt, it's a different set of cards you have for the next experiment. Every time we pick two cards, the deck is changing, so do the probabilities associated with getting two aces.

Consider another random experiment. You would like to predict the weather temperature tomorrow. For this you observe the temperature for an entire year and get the statistical distribution of temperatures. For instance, out of 365 observations you had 50 days with a temperature from 0 to 10 degrees, 100 days with a temperature between 10 and 20 degrees, 150 days with a temperature between 20 and 30 degrees and 65 days with temperature above 30 degrees.Does it mean that a distribution of probabilities of the temperature is as follows?0-10: 50/365 = 13.7%10-20: 100/365 = 27.4%20-30: 150/365 = 41.1%30-99: 65/365 = 17.8%Definitely NOT!The main reason - the Earth rotates around the Sun during the year, that's why in the North hemisphere it's cold in winter and hot in summer. In summer the temperature would be above 20 degrees with a very high probability and in winter it will be below 20 degrees most of the time. So, the conditions of our experiment are changing and, therefore, the results are not applicable for predicting the future.

In summary, if we want to use past results of experiments to evaluate the distribution of probabilities of some random variable, all these experiments must be conducted under identical conditions to assure that we deal with exactly the same distribution of probabilities at all times.If this condition is not satisfied, the validity of our conclusions about the distribution of probabilities of our random variable are quite questionable.

Friday, January 22, 2016

Mathematical Statistics is, in some way, a subject with an inverse purpose relative to Theory of Probabilities.

Theory of Probabilities's purpose is to predict the future behavior based on known distribution of probabilities of a random variable. For example, knowing that the probability of rolling 1 on a dice equals to 1/6, we can predict that out of the next 1000 rolls number 1 will occur, approximately, 1000*(1/6)=167 times. We can even estimate possible deviations from this number.

Mathematical Statistics solves an inverse problem: knowing the results of an experiment in the past, determine the distribution of probabilities of a random variable. For example, if we rolled the dice 1000 times and number 1 occurred 160 times, we can evaluate the probability of this event to be somewhere around 160/1000=0.16.

Why have we decided that the probability of an event can be estimated as the ratio of its occurrence to a total number of experiments? That is based on the theorem proven in Theory of Probabilities as the The Law of Large Numbers with proper evaluation of the precision of such estimate based on the number of experiments.

We can say, therefore, that Mathematical Statistics and Theory of Probabilities are used together with the latter serving as a theoretical foundation of the practical information delivered by the former.

Unfortunately, in many cases people are not concerned about the theoretical foundation and make far fetching statements based on insufficient or wrongly interpreted data. Consider market research, weather forecasting, predictions of the results of presidential elections etc. All these activities are extremely important and that is why it is very important to always understand the limits, the precision and, ultimately, the validity of statistical results.

Here is a short example of the wrong interpretation of statistics. To predict the results of presidential elections in the United States, one company asked 100 people about who, in their opinion, would win - the Democrat or the Republican. It's got 60 responses in favor of Democrat and 40 in favor of Republican. So, it declared the victory for Democrat. Is that right?Another company did the same and from 100 people received 60 responses in favor of Republican and 40 in favor of Democrat. So, it predicted a victory of Republican. Is that right?They cannot be both right!Somewhere there was a mistake. Where? Is there any way we can correct this mistake?

This and many other nuances accompany any statistical research. The purpose of studying Mathematical Statistic in this course is to learn how to do it right.

Before going any further, let's examine a simple probabilistic task. Can we say that the value of a random variable, obtained in a single experiment, is a good estimate of its mathematical expectation?Obviously, it depends on probabilistic properties of our random variable. If it has a variance close to zero, the answer to this question is definitely "yes". However, if the variance of our variable is relatively large, as compared to its mathematical expectation, the answer is "no".

Let's assume that we conduct random experiments to observe the values of random variable ξ. Let's further assume that our experiments are independent and the probabilistic characteristics of our variable are not changing from one experiment to another.Let's say, as a result of N experiments we have obtained values X1, X2...XN of our random variable. What can be done with these values to evaluate certain probabilistic characteristic of random variable ξ?

Here is a simple approach to evaluate the mathematical expectation of ξ.Calculate an average of our N values from a series of N experiments with random variable ξ:M = (X1+X2+...+XN) / NSince each Xi is a value that our random variable ξ took in the i-th experiment, average M can be interpreted as a result of a single combined experiment that occurs when we observe a random variableη = (ξ1+ξ2+...+ξN) / Nwhere each ξi is a random variable distributed exactly as our random variable ξ, and all ξi are independent from each other.

As we see, random variable η has the same mathematical expectation as ξ, but its values are, generally, closer to this expectation since its variance is N times smaller (and, therefore, standard deviation is smaller by √N times).

Thursday, January 21, 2016

We are familiar with polar coordinates on a plane. They are defined by a fixed point of origin (pole), a ray originated at this origin called polar axis and a polar angle or asimuth.For any given point on a plane its polar coordinates consist of two numbers - non-negative distance from the pole called radius and non-negative (counterclockwise) angle from a polar axis to a ray connecting a pole with a given point.

Cylindrical coordinates are an expansion of polar coordinates on two-dimensional reference plane into a third dimension along the longitudinal or cylindrical axis perpendicular to a reference plane at the origin (pole). Traditionally, it is pictured as a horizontal reference plane with polar coordinates on it and a vertical cylindrical axis.

If we project any point A in three-dimensional space onto a reference plane, getting point Ap on this plane, the polar coordinates of this projection (radius ρ and asimuth φ) are the first two cylindrical coordinates of point A.The length of a projection ApA with a sign corresponding to positive or negative direction from Ap to A relative to a direction of longitudinal axis is the third cylindrical coordinate called altitude or height, or z-coordinate (since it's similar to z-coordinate in the Cartesian system).

Thus, three coordinates, radius ρ, asimuth φ and altitude z, are cylindrical coordinates of a point in three-dimensional space.They establish one-to-one correspondence between all points in space and three real numbers, ranging as follows: radius: ρ ≥ 0,non-negative asimuth that is below 2π (radians), any altitude.

1. A side surface of a cylinder of a radius R and height H with lower base lying on a horizontal reference plane with a center at the origin of coordinates can be defined by a system of one equation and two inequalities as follows:ρ = Rz ≥ 0z ≤ H

2. A plane going through a vertical (longitudinal) Z-axis and intersecting a horizontal reference plane at a line making an angle Φ with a polar axis on it can be defined by a very simple equationφ = Φ

3. A side surface of a cone of a radius R and height H with lower base lying on a horizontal reference plane with a center at the origin of coordinates can be defined by a system of one equation and two inequalities as follows:z = H·(1 − ρ/R)ρ ≥ 0ρ ≤ R

Spherical coordinates is yet another way to identify a position of a point in three-dimensional space using three characteristics.

In Cartesian system of coordinates we use three linear dimensions to accomplish this (x, y, z), in cylindrical system we use two linear and one angular dimensions (ρ, φ, z), in spherical system of coordinates we use two angular and one linear dimension (φ, θ, r).

Here is how a spherical system of coordinates looks in a three-dimensional space relatively to Cartesian system:

The first characteristic is radial distance from the origin of coordinates (radius) r.

The Z-axis defines zenith direction and angle φ characterizes the deviation of the direction towards our point in space from zenith. It is called polar angle - the second (angular) characteristic in spherical system of coordinates.

The XY-plane is a plane of reference (similar to cylindrical coordinates) with X-axis serving as polar axis. The deviation from polar axis towards a projection of our point onto a reference plane is azimuth θ - the third (angular) characteristic in spherical system of coordinates.

Speaking about traditional letter designation, it must be noted that sometimes the letters φ and θ are used in opposite sense than is described above - letter θ designates a polar angle, while letter φ designates the asimuth.Also, the polar distance sometimes is designated by a letter ρ.

Let's summarize the requirements of the spherical system of coordinates. We need a fixed point of origin O, an axis going through it that defines zenith direction (Z-axis on a picture above), from which we measure polar angle φ, a reference plane going through the origin perpendicular to zenith with an axis of azimuth=0 on it (XY-plane on a picture above with X-axis identifying azimuth of 0, Y-axis is not needed in this system of coordinates), from which we measure asimuth θ, and a unit of linear measurement to measure radial distance.

Given all this, to find the spherical coordinates of a point A in three-dimensional space, we have to make the following constructions and measurements:(a) measure the radial distance r from the origin O to point A along ray OA;(b) construct a plane through Z-axis and line OA and measure within this plane a polar angle φ of deviation from Z-axis, that is an angle from the positive direction of Z-axis to ray OA (it's the smaller angle that is used as a polar angle);(c) project point A onto a reference plane into point Ap;(d) within a reference plane measure an azimuth θ from the positive direction of the reference axis (X-axis on the picture above) counterclockwise to ray OAp.

The inversed procedure to find a point by its spherical coordinates is as follows:(a) the radial distance r defines a sphere our point is located at;(b) the polar angle φ defines a cone, on the surface of which our point is located; this cone intersects a sphere mentioned above at a circle;(c) project the circle obtained above to a reference plane, also getting a circle of the same radius on this plane and, using azimuth θ, find a point on this circle - a projection of a point we have to find;(d) from the projection point obtained above, going along the perpendicular to a reference plane at that point, we go to an intersection with a sphere and a cone constructed above to get the point we need.

Examples of usage of spherical coordinates.

1. A sphere of a radius R centered at the origin of coordinates is defined by a very simple equation with only radial distance r participates:r = R

2. A conical surface with an apex at the origin of coordinates and an angle α between its generatrix and main axis of symmetry can be defined by a simple equation for polar angle φ: φ = α

3. Position of stars relatively to a position on the Earth can be expressed in spherical coordinates with zenith being on the axis of rotation of our planet, reference plane would go perpendicular to it on a parallel of an observer, almost constant polar angle would be a measure of deviation of the direction to a star from zenith and azimuth (deviation from some chosen direction, say, to the North on the reference plane) would change as our planet rotates around its axis.A different specification of the direction to a particular star might be a time and an angle above the horizon. It's more practical, but at the core of it lies the same spherical system because the time is, actually, a measure of the Earth' rotation, that is, a replacement for azimuth, and an angle above horizon is a replacement for a polar angle.

This lecture is about geographical coordinates - the system of coordinates on a surface of our spherical (for all intents and purposes) planet Earth.Our purpose is to numerically identify a position of any point on Earth - a very useful thing in many practical cases.There are more than one way to accomplish this, but we will talk only about one particular method, standard across many areas of activity, based on longitude and latitude.

Let's assume that we have a sphere that represents our planet Earth (we will use terms "sphere" and "Earth" interchangeably) and a point on it. We assume that our sphere is rotating around some axis that connects North Pole and South Pole. We also assume that direction from each point on this sphere that coincides with the direction of rotation is called East and the opposite direction is called West. The pole to the left from a point while looking in the direction of rotation is the North Pole, the other is the South Pole.

We can identify the position of any point on this sphere with two numbers - both angular measures - using the following definitions.

1. Let's define a concept of a meridian going through some point P on a sphere.

Meridian is a semi-circle on a surface of a sphere, going from North Pole to South Pole through our point, which is an intersection of a sphere with a half-plane going through an axis of rotation of the Earth from North Pole to South Pole and point P.

There is one and only one meridian that goes through any point on a sphere, except the two poles - just draw a one and only one half-plane through this point and an axis of rotation of the Earth, this half-plane's intersection with a surface of the Earth is a semi-circle going through our point from pole to pole, that is the meridian of this point.

To identify a specific meridian, we choose an angular measure of a dihedral angle between a half-plane generating this meridian and some fixed half-plane generating some base (fixed) meridian. For historical reasons the fixed half-plane serving as the base for this measurement is the one generating a meridian that goes through Royal Observatory in Greenwich - now a district of London. This Greenwich meridian is called Prime Meridian and is characterized by a dihedral angle of 0o.From Prime Meridian all those towards East will have a measure of corresponding dihedral angle from 0o to +180o (or 180oE). Those towards West will have measures from 0o to −180o (or 180oW).This angular measure of deviation of a meridian of some point from Prime Meridian is called longitude and is denoted by letter λ on the picture above. Knowing the longitude of a point on a sphere, we can construct its meridian - a curve on which this point is located.

2. To identify a position of a point on its meridian we will introduce a concept of a parallel and another measure - its latitude.

Parallel is a circle on our sphere formed by an intersection of any plane perpendicular to a sphere's axis with a sphere's surface. The parallel formed by a plane cutting the axis in the middle (that is, going through a center of a sphere) is called an equator.

The position of any parallel on a sphere is fully defined by an angle between an equatorial plane and a radius from a sphere's center to any point on this parallel (angle φ on the picture above). The measure of this angle is called latitude. It ranges from 0o for any point on the equator to 90o for both poles. Points closer to North Pole than equator will have a designation N with their latitude, points towards South will be designated with letter S.

As we see, for any point on a sphere we can determine its longitude (a specific meridian it is on) and latitude (a specific parallel it is on). Combination of these two curves, a meridian and a parallel, on a surface of a sphere fully defines the position of a point.

Every point on a sphere, except two poles, uniquely defines its coordinates, longitude and latitude. In case of North Pole, it is reasonable to attribute only latitude of 90oN to it. Similarly, for South Pole it's 90oS.

To more precisely identify a position, smaller than 1o units of angle measurement are used: angular minute (1') equals to 1/60 of 1o;angular second (1") equals to 1/60 of 1'.Alternatively, decimal fractions of 1o can be used as well.

In addition, it should noted that in reality our planet is not a perfect sphere and precise definitions of longitude and latitude are slightly different from those explained above.Also, for those interested in coordinates above the surface of the Earth, they can add a height above the surface as a third coordinate. So, an object above the ground will have longitude λ and latitude φ of its projection on the ground plus height h above that point.

Tuesday, January 12, 2016

The purpose of having any coordinates in space is to represent a
position of a point numerically as a sequence of real numbers. In
three-dimensional space this requires a sequence of three real numbers.

Using this representation, we can describe relationship among many
points in space (say, points lying on some plane or on a surface of some
sphere) algebraically.

This approach is fully utilized in a subject of Analytical Geometry that is studied in universities.

Cartesian Coordinates in Space

Cartesian coordinates in space are similar to Cartesian coordinates on a plane and just need an extension to a third dimension.

In any system of coordinates, first of all, we need a point of origin O - some fixed point in space relatively to which we describe a position of any other point.

Next let's define axes (plural of axis) of coordinates.

I think, it's more natural to define first an axis that differentiates
three-dimensional geometry from two-dimensional one on a plane. So,
let's choose a line going through the point of origin O, call it Z-axis and choose a particular direction on this line as positive.

We can conditionally call this Z-axis a vertical axis because it is usually drawn vertically on pictures and illustrations.

Consider now a plane going through the point of origin O perpendicular to Z-axis. This plane will contain two other axes - X-axis and Y-axis - and we will call this plane XY-plane.
We can choose X-axis first within this plane as we wish, then Y-axis
would be a line within this same plane and perpendicular to X-axis.

Both X-axis and Y-axis require a positive direction. Traditionally, we
choose a direction of X-axis as we wish and then choose a direction of
Y-axis in such a way that, looking from the positive ray of Z-axis,
positive ray of X-axis can be rotated counterclockwise to coincide with a
positive direction of Y-axis.

Similar construction could be done if we start from XY-plane, choose
X-axis and Y-axis within it and construct Z-axis as a perpendicular to
XY-plane.

Alternatively, we can start with only X-axis, then choose Y-axis
perpendicular to it, thus defining XY-plane, and then construct Z-axis.

Similar to XY-plane, we can define XZ-plane that contains X-axis and Z-axis or YZ-plane that contains Y-axis and Z-axis.

All these methods lead to similar result of having three axes of coordinates, X-axis (denoted as x), Y-axis (denoted as y and Z-axis (denoted as z), mutually perpendicular to each other.

Finally, we have to choose a unit of measurement to be able to convert
geometrical position of any point in space into three coordinates
relative to three axes defined above.

Assume now that we have a point A somewhere in space,
where three axes of coordinates are defined as described above. Our task
is to convert a geometrical position of this point to a sequence of
three real numbers that uniquely represent it. In other words, we will
put into one-to-one correspondence a set of all points in
three-dimensional space to a set of all triplets of real numbers.

The simple way to do it is to draw three plains through this point A, each perpendicular to a corresponding axis (a unique construction):

plain α⊥axis x (let α∩x=Ax)

plain β⊥axis y (let β∩y=Ay)

plain γ⊥axis z (let γ∩z=Az)

The above defines three projections of point A onto each axis - Ax, Ay and Az, uniquely defined by a chosen position of point A.

Having a unit of measurement, we can associate the length of segments OAx, OAy and OAz
with real numbers - their length in chosen unit, and assign a sign to
this number - positive if corresponding projection point lies towards
the positive direction of a corresponding axis or negative in an
opposite case.

The above procedure associates a triplet of real numbers with a position of a point in three-dimensional space.

These three numbers have proper names:

signed length of segment OAx is called x-coordinate or abscissa;

signed length of segment OAy is called y-coordinate or ordinate;

signed length of segment OAz is called z-coordinate or applicate.

Let's do an inverse operation to associate a point with a triplet of real numbers.

Obviously, three real numbers can be easily converted into three points
on our three axes by constructing three segments along them - OAx, OAy and OAz. Then we construct three planes through points Ax, Ay and Az correspondingly perpendicular to axes they belong to.

These three planes intersect in one and only one point A, which is the point associated with given triplet of real numbers.

It would be a useful exercise to prove that our relationship between
points in three-dimensional space and triplets of real numbers is indeed
one-to-one correspondence. Rigorous proof of this is very much related
with foundations of geometry and certain axioms that are beyond the
scope of this course. Intuitively, however, it is as obvious as the fact
that for each real number r we can find a point R on a straight line that is on a distance equal to this number (using some unit of measurement) from a certain fixed point O, that is OR=r (for positive r point R is on one side from point O, for negative R - on the other).

As an exercise, here are a few examples of using coordinates to express the geometrical objects and properties.

1. Origin of coordinates:

Point O(0,0,0).

In a form of equations, it's a system of three equations

x = 0

y = 0

z = 0

2. Equation that describes all points on XY-plane:

For any point on XY-plane a plane perpendicular to Z-axis is that same XY-plane.

It intersects Z-axis at an origin of coordinates, so its Z-coordinate must be equal to zero.

So, the equation of XY-plane is

z = 0

3. Equation of an angle bisector between Y-axis and Z-axis:

Since all points within YZ-plane are characterized by an equation x = 0 and within YZ-plane points lying on an angle bisector between Y-axis and Z-axis are characterized by a property y = z, the final numerical representation of this line is a system of two linear equations

x = 0

y = z

4. Equation describing points on a sphere of a radius R with a center at the origin of coordinates:

The square of a distance from the origin of coordinates to a point A with coordinates (x,y,z) is, as follows from Pythagorean Theorem applied twice, equals to x²+y²+z². Therefore, an equation for a sphere of radius R is

Thursday, January 7, 2016

Problem AFind the radius r of a sphere inscribed into a right circular cone of altitude H and radius of a circular base R.

Answerr = R·H ⁄ [R+√(R²+H²)]

Problem BTwo vertices at the base of an isosceles triangle lie on a circle at the base of a right circular cylinder, while the third vertex of this isosceles triangle lies on a cylinder's side surface.The length of the base of a triangle is a=6, its altitude is h=2, angle between a plane where our triangle is located and the plane of the base of a cylinder is β=30o.What is the radius R of a cylinder?

Tuesday, January 5, 2016

ProblemConsider our planet Earth to be an ideal sphere of a radius R.Let's circumscribe a right circular cone around it with its apex lying on the continuation of the Earth's axis of rotation somewhere above the North Pole and its base tangential to the Earth's surface at the South Pole. So, the Earth's axis is an altitude of this cone.Each generatrix of the cone's side surface is a tangent to the surface of the Earth, so all points of tangency form some curve (prove that this curve is a circle lying in the plane perpendicular to the Earth' axis).The circle of tangency between the Earth and a side surface of a cone is the 60th parallel of the North hemisphere (going through Alaska, Norway and other places). Each of its points is located at an angle 60o from the Earth's center relatively to a horizon, that is having 60o latitude.What is the volume of a cone?

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